\(\int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx\) [924]

Optimal result

Integrand size = 31, antiderivative size = 59 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \]

[Out]

hypergeom([1, 1+m],[2+m],-d*(1+sin(f*x+e))/(c-d))*(a+a*sin(f*x+e))^(1+m)/a/(c-d)/f/(1+m)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2912, 70} \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {(a \sin (e+f x)+a)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)} \]

[In]

Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x]),x]

[Out]

(Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d
)*f*(1 + m))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+x)^m}{c+\frac {d x}{a}} \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \]

[In]

Integrate[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x]),x]

[Out]

(Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d
)*f*(1 + m))

Maple [F]

\[\int \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{c +d \sin \left (f x +e \right )}d x\]

[In]

int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x)

[Out]

int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x)

Fricas [F]

\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \]

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m/(c+d*sin(f*x+e)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \]

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)

Giac [F]

\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \]

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c+d\,\sin \left (e+f\,x\right )} \,d x \]

[In]

int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x)),x)

[Out]

int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x)), x)